An algorithm for computing genera of
ternary and quaternary quadratic forms
Rainer Schulze-Pillot
Fakultiit f. Mathematik
SFB 343, U niversitiit Bielefeld
D-4800 Bielefeld
Introduction
This is a preliminary report on an algorithm for computing genera of ternary and quaternary positive definite
quadratic forms over l.
It is well known that due to the simple shape of the reduction conditions in these dimensions [Mil, Ca] it is
in principle no problem to compute representatives of all classes of such quadratic forms whose discriminant
is below a given bound. This has been done by hand in the ternary case for half- discriminant up to 1000 by
Brandt and Intrau [BI], for quaternaries see [Ge, To]. It is, however, sometimes desirable to be able to quickly
determine representatives of all classes in some fixed genus of quadratic forms of possibly high discriminant
without having to generate along the way all forms of smaller discriminant.
An obvious attempt in such a case is to use Kneser's method of neighbouring or adjacent lattices [Kn1] that
has proved to be useful for the construction of unimodular lattices in "middle" dimensions like 24 and 32 [Ni,
CS, KV]. This report intends to draw attention to the fact that it is indeed not difficult to use this method
in dimensions 3 and 4 as the basis of an algorithm that serves our purpose. With almost no extra work one
obtains at the same time the "adjacency graph" of the classes determined; this has interesting arithmetic and
graph theoretic (see Sect. 1) applications.
In Section 1 we collect some basic facts and notations from the theory of quadratic forms and explain the
method of neighbouring lattices.
The very simple algorithm based on this method is described in Section 2 where one also sees which difficulties
arise in higher dimensions. We intend to use our algorithm for the experimental investigation of the Fourier and
Fourier-Jacobi coefficients of certain linear combinations of Siegel theta series of quaternary quadratic forms;
this is still in progress. The underlying problems from the theory of modular forms which were our starting
point for this project are briefly sketched in Section 3.
The C-programs for the implementation of our algorithm are not included, they can be obtained from the
author. I want to thank Britta Habdank who, for the ternary case, translated into C and completed a part
of the program which I had originally written in Pascal, Dirk Tubbesing who wrote the program for unique
reduction in dimension 3 after Dickson [Di] and a test for equivalence in the quaternary case, and R. Scharlau
who supported these works within his project on constructive methods in the theory of quadratic forms.
1 Neighbouring lattices
Let V be an m-dimensional vector space over Q, let Q : V _,. Q a non-degenerate quadratic form with associated
bilinear form
B(x, y) = 1/2(Q(x + y)- Q(x)- Q(y)),
L = le1 + ... + lem a l-lattice on V of full rank and assume B(L, L)
it is even (type II) if Q(L) ~ 2l, odd (type I) otherwise.
~
l. The lattice Lis then called integral,
Two lattices L, L' are isometric (or in the same class) if there is a linear isomorphism from L onto L' preserving
Q, they belong to the same genus if their p-adic completions Lp, L~ are isometric for all p. For the notion of
spinor genus see [OM, Sect. 102].
Let A = (B( e;, ej)) be the Gram matrix of Q relative to the basis e1, ... , em of L, put d( L) = det A. If Q has
Gram matrix B with respect to some basis e~, ... , e~ of some lattice L' on V then L and L' are isometric if
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and only if there is T E G Lm (l) with T' AT = B, in which case A and B are called equivalent.
The dual lattice of L is £# = {x E VIB(x, L) ~ Z}, one has (£# : L) = d(L). One calls L unimodular or
self-dual if£# = L.
M. Kneser introduced in [Knl] the following method for constructing integral lattices L' on V with d(L') = d(L):
Let p f d(L) be a prime, x E L with 112 IQ(x) and p- 1 x rt L. Put Lx = {y E LIB(x, y) E pZ} = L n (Zi + £)#,
put L' =
z~
p
+ Lx.
L' is then called a p-neighbour of(p-adjacent to) Land one has
(i) d(L') = d(L)
{ii) L' is integral
{iii) L' is even if and only if Lx is even and Q(x) E: 2p2Zp, i.e., if and only if either Lis even and Q(x) E 2p 2lp
or L is odd, p = 2 and Lx is equal to the evE~n sublattice L 0 := {y E LIQ(y) E 2Z} of L.
{iv) L and L' are in the same genus if and only if they are either both even or both odd.
These properties are well known, see [OM, Sect. lOG] and [Knl]. Properties i)-iii) are immediate consequence;;
of the fact that one has B(x, L) 2 d(L)l for primitive x (which follows from (L# : L) = d(L)) and hence
(L: Lx) = p = (L': Lx) holds. Property iv) follows from i)-iii) since L' differs from L only in the completion
at p and a unimodular lattice over lp is determined (up to isometry) by dimension, discriminant, and type (see
e.g. [CS,Ch. 15]).
We list some further properties of the construction. These are certainly not new, but the available reference>
list only small portions of them.
(v) Ifp
II d(L)
then i)- iv) are also true.
This is true because one can then write Lp = lpz .l. J{ with Q(z) E pl; and unimodular K. If one had Lx = L,
then x = az + x' with x' E f{ divisible by p and Q(x) E p 2lp would imply a E plp which contradicts p- 1 x rt L.
As above this shows i) - iii). iv) is dear if either L and L' are both even or p =f:. 2 since in that case Lp and
L~ are Zp-maximallattices in Eichler's sense and hence [E2, Satz 9.6] isometric; in the remaining case one can
check it with the help of [CS, Ch. 15, Th. 10].
(vi) Let x1, x2 E L be as above, L 1 and L 2 the respective neighbouring lattices. Then for p =f:. 2 one has L1 = L2
if and only if lx1 + pL = lx2 +pL. For p =: 2 one certainly has L1 = L2 if lx1 + 4L = Zx 2 + 4L. More
precisely, if p = 2 and L is even, then L1 == L2 if and only if Q(xl) ::: Q(x2) mod 8 and lx1 + 2L =
lx2 + 2L. If p = 2 and L is odd, assume x 1 and x 2 to be such that gcd(Q(x;), 8);:::: gcd(Q(x'), 8) for all
x' E Zxi + 2L 0 with x' 2L (i = 1, 2).
rt
Then again L1
this way.
= L2 if and only iflx1 + 2L'l = Zx 2 + 2£0, and all neighbours L' of L can be obtained in
The assertions for p =f:. 2 and for even L are obvious. For the rest, fix some x E L and consider x' = ax + 2z for
z E L, a odd. If z E L 0 then Q(x')
Q(x) +4B(x,. z) mod 8, whereas for z E Lx one has Q(x'):::: Q(x) +4Q(z)
mod 8. This shows that Q(x')::: Q(:e) mod 8 for all x' E Zx + 2L 0 (or x' E lx + 2Lx), x' rt 2L, if and only if
Lx = L 0 • We notice next that obviously (x1, x2 arbitrary) L 1 = L 2 if and only if Zx 1 + 2Lx, = lx2 + 2L:c,·
Thus if L' = l ~ + Lx is a neighbour of L and gcd( q(x ), 8) is not maximal in lx + 2L 0 , then L 0 =f:. Lx and we can
replace x by x' = x + 2z with z E Lx, z ~ L 0 satisfying 8jQ{x'), and we see that indeed all neighbours of L can
be constructed using a vector x with gcd(Q(x), 8) maximal in Zx+2L 0 • If now x 1, x 2 are such that gcd(Q(x;), 8)
is maximal in lx; + 2L 0 (i = 1, 2), assume first L1 = L 2. Then x 1 = ax2 + 2z with a odd, z E Lx,· If z rt L 0 ,
then Lx, = Lx, =f:. L 0 and the maximality of X], x 2 implies 8IQ(x;) (i = 1, 2), in contradiction to z ~ L 0 • Hence
z E Lxo n L 0 and therefore Zx 1 + 2£0 = lx2 + 2L 0 • If we assume on the other hand Zx 1 + 2L 0 = lx 2 + 2L 0
we conclude in the same way lx1 + :~Lx, = Zx2 + 2Lx 2 , hence L1 = L2.
=
(vii) If p is odd and p f d(L) then lx + pL conJ:ains p-primitive vectors x' with Q(x') E p 2lp if and only if
Q(x) E pl. One finds such an x 1 by choosing y E L with B(x, y) = a f/_ pl (e.g. a suitable vector of an
mod p and putting x' = x + apy. If p = 2
arbitrary basis of L), solving the congruence 2aa ::: ~
p
then lx 2L 0 contains 2-primitive vectors x' with Q( x') E 4Z if and only if Q( x) E 4Z. If Q( x) E 4Z
+
135
and Lx :/; L0 one finds x' E lx
x' = x ± 2y.
+ 2L
0
with 8IQ(x') by choosing y E U with B(x, y) odd and putting
The two final points are concerned with the question which classes in the genus of L are obtained by successive
construction of neighbours.
(viii) By successive construction of p-neighbouring lattices one obtains all lattices on V of the same discriminant
as L which lie in l[~]L. For p :/; 2 these lattices are all in the same genus as L. For p = 2 and even L
one obtains all lattices in the genus of L that are in l(!]L if one constructs neighbours only with vectors
x satisfying Q(x) E 81.
To see this let K ~ l[~]L with d(K) = d(L), let (L: L n K) = pr = (K: L n K) (the indices are equal because
of d(L) = d(K)). Choose x' E K, x' ~ L, then p'x' =:xis p-primitive for somes EN (since by the elementary
divisor theorem one has aK ~ L for some a E N). Let L' =
+ Lx be the neighbour of L constructed from
x, then L' n K ~ L n K since p•- 1 x' = !. is in L' n K but not in L n K. Hence (L' : L' n K) < (L : L n K),
and iterating this one can connect L ana K by a chain of lattices p-neighbouring to each other. Obviously,
if p = 2 and L and K are even then the same is true for all lattices in the chain and all the vectors used in
the construction have length divisible by 8. As noted in iv) the construction of neighbours does not leave the
genus except if p = 2 and one changes the parity (type) of the lattice. We note at this occasion that K can be
constructed as a neighbouring lattice of L if and only if (L : L n K)
(K : K n L) p, in particular being
neighbours is a symmetric relation.
(ix) If the genus of L consists only of one spinor genus {OM, Sect. 102], the rank m is at least 3 and the
completion vp is isotropic (i.e., there is 0 :/; y E vp with Q(y) = 0) then successive construction of
li
=
=
p-neighbouring lattices yields representatives of all classes of lattices in the genus of L.
If p = 2 one can also obtain representatives of the (unique) genus of L containing the 2-neighbours of L
whose type (parity) is different from that of L.
The condition that the genus of L should contain only one spinor genus is in particular satisfied if no odd
prime divides d(L) to the power m(~-l) and if 2 does not divide d(L) to the power
{
(3m-2J(m-1)
ifm is odd
m(3m-5)
ifm is even.
2
Our general condition p 2 f d(L) implies that vp is isotropic for odd p, if or m
or m 2: 5. If m = 3 and p is odd, vp is certainly isotropic if p f d(L).
2: 4,
for p = 2 if L is even
The first part of ix) can be found in [Knl) for p = 2, it follows in the same way for arbitrary p. One should notice
that it depends on the strong approximation theorem [Kn2, El) which can be applied only if Vp is isotropic.
The bounds for the power, to which primes may divide d(L) are from [Kn3) (odd p) and [EH) (p = 2). The
remarks on the isotropy of vp can easily be checked using the classification of unimodular p-adic lattices [OM,
Sect. 92, 93].
If the genus of L happens to contain more than one spinor genus, one can reach (starting from L) the other
spinor genera in the genus with the neighbouring lattice construction by carefully selecting some special primes
and constructing an arbitrary neighbour of L at each of these primes. Successive construction of p-neighbours
of all these lattices for some arbitrary p with p 2 f d(L) then gives all classes in the genus of L. We do not go
into details here but refer to [BH).
To finish this section we remark that by viii) we can use our construction to define a graph whose vertices are
the lattices in l[~]L of the same discriminant as L (or in the same genus as L), with two lattices joined by an
edge if they are neighbours of each other. Form= 3 and p = 2 or p f d(L) this graph is a tree and has been
studied in [SPl). If one identifies isometric lattices and introduces loops and multiple edges, one obtains a finite
multigraph, where the multiplicities of the edges are given by Eichler's Anzahlmatrices [E2). The multigraph
obtained for m = 3 is closely connected with the "Ramanujan graphs" which recently found much interest in
136
connection with applications in combinatorics and network theory [Pi2, LPS]. We plan to deal with this aspect
of the neighbouring lattice construction in another place.
2 The algorithm
We are now going to describe our algorithm, doing this as far as possible for general m 2: 3. We assume that
the genus of L has only one spinor genus and that p "I 2 or L is even. The necessary modifications for p = 2
and L odd can be performed using vil), vii) of Sect. 1 and are left to the reader. We first choose a prime p with
p2 f d( L), if m = 3 and p is odd we require (because of ix)) even p f d( L).
The algorithm has the following main steps (which we will analyze more closely below):
Algorithm (Construct forms in a fixed genus)
Input: An integral positive definite symmetric m :x: m-matrix A (m 2: 3) {Gram-matrix of a lattice Las above)
and a prime p as above
Output: A list of Gram matrices of a set of representatives of the classes in the genus of (lm, x' Ax).
Step 1: Replace A by an equivalent reduced matrix, open a list of matrices with A as first element
Step 2: Put (L, Q) = (lm, x' Ax), find the classes lx
Q( x) E 2p 2 lp and in each class one such x.
+ pL
in LfpL which contain p-primitive x E L with
Step 3: For each x from step 2 determine a reduced Gram-matrix Bx of the neighbouring lattice
mark A.
l~
+ Lx;
Step 4: For each Bx from step 3 compare Bx with the matrices in the list. If Bx is equivalent to a matrix
already in the list, take the next Bx. If not, add Bx to the list.
Step 5: If the list contains no unmarked element, terminate. Otherwise let C be the first unmarked matrix,
put A <- C, go to step 2.
Of course these steps need some explanation.
Steps 1 and 5: The bookkeeping for the list is conveniently organized using pointers. It turns out to be useful
to organize it in two different ways:
Once with pointers pointing to the matrix that has been generated next (this is the order used in step 5), once
ordered by some concept of size (e.g., lexicographic order of the diagonal) (this order is useful for the comparison
operations of step 4). For the meaning of "reduced" see step 4.
Step 2: This is done using vii) of Sect. 1 by going through the lines in L/pL (L/2L 0 for p = 2). If such a
(nonzero) line is represented by x' with Q(x') E pl (Q(x') E 4Z for p = 2) one can replace it by x representing
the same line and satisfying Q( x) E p 2 l ( Q( x) E SZ for p = 2) by the procedure given in vii). The required
vector y E L with B(x', y) 1, pl can be chosen to be one of the basis vectors of L; in case p "I 2 and pld(L) it
can happen that B( x', L) <;;; pl, in this case there is no primitive vector x E lx' + pL with Q( x) E p2l. Since
the number of lines "I 0 in LfpL is (pm - 1)/(p ·- 1) this procedure obviously becomes impractical for large
m. If L happens to have a large group of automorphisms one can reduce the work by running only through
a set of representatives of the orbits under this ~;roup. However, it is firstly difficult to control this a priori
(without precise knowledge of the lattice and its group) in an algorithmic way, and secondly, in some sense
"most" lattices have trivial automorphism group [Bi, Ba]. For this reason the neighbouring lattice method has
been used so far for large m only for clever construction by hand of all (in the case of big automorphism group)
or some neighbouring lattices.
The number of classes lx + pL that are suitable for the method is in case p f d(L) or p = 2 and m odd given
by [Kn4, Sect. 12]:
(pm-1
{
+ pm/2 _ pm/2-1 _
1)/{p _ 1)
m even,
( (-1)~
+ pm/2-1 _
1)/(p _ 1)
m even,
(
(pm-1 _ pm/2
(pm-1 - 1)/(p- 1)
2
d(L)) __ 1
(-l)m:2d(£)) --
-1
modd
and is hence of the order pm- 2 •
Step 3: Here we have the task of determining a basis {e~ =
m
L: tjiCj}
of the neighbouring lattice l ~
+ Lx
({Ci}
j::;l
a basis of L ), this gives then Bx = T' AT. This basis can be determined using the modified LLL-algorithm
137
(M LLL) [PZ, p. 209]: One first finds a basis of Z~ + L, the dual basis of this is a basis of (Z~ + L)#. Using
the elementary divisor algorithm[PZ, 180ft] one finds a basis of L~: = L n (Z~ + L)# and another application
of M LLL gives the required basis of L = z~p + Lz. In the case p = 2, L even (which we have implemented)
one can proceed more directly. Following the procedure from vi), vii) of Sect. 1 the vectors x found in step 2
all have at least one coordinate = ±1, hence can be exchanged for one of the basis vectors. If the new basis of
Lis {e 1, ... ,em_ 1,x}, then {e1, ... ,em-1, His a basis ofZ~ +L, and one has L' = Z~ EBK where
K = {y E Ze1
+ ... + lem-1IB(x, y) E 2Z}.
A basis of K can be determined by a simple distinction of cases, taking account of the parities of the B(x, ei)
(i = 1, ... , m- 1), which gives the desired basis of L'.
Step 4: This step contains the main practical difficulty in the algorithm because it is in general quite time
consuming to decide whether two given Gram matrices A and Bare equivalent. In dimensions 3 and 4 we can
improve on the general method of [Poh] by making use of reduction theory. Let us describe our method in more
detail:
Recall that a basis {e1, ... , em} of L is called (Minkowski-) reduced if one has for 1 ~ j ~ m:
m
Q(e;)
= min{Q(y)ly= l:aiei with gcd(a;, ... ,am) = 1}
i=l
[Mi2, vdW, Ca], the corresponding Gram matrix is then also called Minkowski reduced. For m :::; 4 one has in
addition [Mil, Ca]:
m
If { e1 , ... , em} is not reduced then there exists y :::
L: aiei with ai E {0, ±1} and Q(y) < Q( ei) for some i with
i=l
ai = ±1. That is, by running through all vectors with coordinates 0, ±1 one can transform any given basis into
a Minkowski reduced basis in finitely many steps.
This gives first the following
Algorithm (Minkowski-reduction)
Input: A positive definite integral symmetric (m x m)-matrix A (m ~ 4)
Output: A Minkowski reduced matrix equivalent to A.
Step 1: For j = m, ... , 1: (put a; = 1, aj+l• ... , am = 0,
for a;-1::: -1,0, 1
for a1 = -1,0, 1
m
(put y
= l:aiei,
i=l
if Q(y) < Q( e;) then
(T = Em,tii- Yi (i = 1, ... ,m),
A - T' AT, go to step 1)))
Step 2: Terminate, output A.
Observe that Minkowski reduction is not unique, i.e., we can have Minkowski reduced matrices A =/: B which
are equivalent. However [vdW, Ca], if A, B are Minkowski reduced in dimension~ 4, the diagonal coefficients
are equal to the successive minima and hence aii =/: bii for some i implies that A and B are not equivalent. For
m = 3 one can prescribe additional constraints on the off-diagonal coefficients that make reduction unique; such
constraints have been given by Dickson [Di, p .... ],who at the same time shows how to obtain this unique normal
form from a given Minkowski reduced matrix. Using Dickson's method our test for equivalence in dimension 3
is then simply a test for equality of the Dickson reduced matrices.
138
In dimension 4 one does not seem to know such a unique normal form which can be determined in a straightforward algorithmic way, e.g. by exchanging one basis vector at a time as we did for Minkowski reduction; some
criteria for inequivalence and some examples of equivalent Minkowski reduced matrices can be found in (Ge].
We tried the following method to reduce the matrix further:
Algorithm (off diagonal reduction)
Input: A Minkowski reduced 4 x 4--matrix A
Output: A matrix equivalent to A with small sum of the absolute values of the off diagonal coefficients, small
maximal absolute value of the off diiagonal coefficients, nonnegative last row, minus signs as far left and up as
possible.
Step 1: For j = 4, ... , 1
{ CXj +-
For
1, a.;+l
0, ... , CXm
+-
0;
== -1,0, 1
CXj-l
For a 1
+-
= -·1, 0,1
4
{y
+-
I:O;e;;
i=l
If Q(y) = Q(ej) and
(t,
(t,
IB(y, ei)l <
i¢j
or
t,
t,
ia;il
i¢j
IB(y, e;)i =
i¢j
and
ia;j I
i¢j
~fiB(y, e;)l < rp'#flaij 1))
{T +- E4;t;i
+-
y; (i = 1, ... ,4);
A+- T' AT; go to step 1}} }.
Step 2: Order A by the following critera (in order of their priority, by changing the order of the basis vectors
of L).
a) i ~ j
=> a;;
~
aii
4
b) i ~ j
=>
L
k=l
k¢i
4
ia;ki::;
L iaikl
k=l
k¢j
c) Fork=4, ... ,2: Ifi::;j, laiil=laidfor/:=k+1, ... ,4,thenla;kl::;laikl
Step 3: Order the signs in A by the following criteria (in order of their priority, by setting e;
basis vectors e; of Land by changing the order of the basis vectors e; of L):
+-
-e; for some
a) a;4;:::: 0 (i = 1, ... , 3)
b) a;i ;:::: 0 (i = 1, ... , 4) if a; ,i +1 = ... = a;4 = 0 (j = 3, ... , 1)
c) a23 ~ 0 (if this is possible without violating 2 or 3a), 3b)).
Step 4: Output A
The reduction obtained seems to be remarkably dose to being unique. For example, we used our algorithm to
compute all classes in the genus of quaternary quadratic forms of discriminant 389 2 and level 389; this genus has
319 classes by known results from the theory of quaternion algebras. Using the reduction described above and
a test for equality of the reduced matrices as equivalence test the algorithm produced only 10 Gram matrices
that were in fact equivalent to other matrices in the list.
139
On the basis of our experience we propose hence the following test for equivalence in dimension 4 (which worked
without excessive use of computing time in our examples):
Algorithm (test for equivalence)
Input: Two positive definite integral symmetric (4 x 4)-matrices
Output: eq or ineq
Step 1: Reduce A and B by Minkowski reduction
Step 2: If a;; =f. b;; for some i terminate, output ineq
Step 3: Reduce A and B further using off diagonal reduction
Step 4: If A= B terminate, output eq
Step 5: Test for equivalence by brute force, i.e., by searching for vectors Xl, ... ,X4 E Z4 with xiAxj = bij·
The vectors x E Z4 with x' Ax ~ b44 can be found relatively fast using (e.g.) the algorithm from [PZ, p. 190]
(this has been programmed for our purpose by Dirk Tubbesing)
It is not difficult to see that our algorithm produces large numbers of equivalent matrices. Most of these
already become equal after our reduction procedure and are therefore caught in step 4. On the other hand only
a relatively small percentage of the inequivalent matrices agrees in the diagonal coefficients after Minkowski
reduction, so that inequivalence is usually detected in step 2. Thus, the relatively time consuming step 5 is not
invoked too often.
In further experiments it may prove useful to add between step 4 and 5 one or more of the following inequivalence
tests:
< a33 ,
If a33 < a44,
a) If a 22
test the upper left 2
X
2 blocks of A and B for equivalence
b)
test the upper left 3
X
3 blocks of A and B for equivalence
c) Determine the level N of the genus (the common denominator of the entries of A- 1 for any A in the
genus), test whether
r(A, n) =f. r(B, n) #{x E l 4lx'Bx n}
=
for some n
<
=
iN [J (1 + 1).
PIN
p
a) and b) have been proposed in [Ge], if they give inequivalence, A and B are certainly inequivalent. Also
r(A, n) =f. r(B, n) for some n in step c) certainly proves inequivalence. If these representation numbers are equal
for all n <
[J (1 + 1 ), then by [He, p. 811] the theta series of degree 1 of both quadratic forms agree. This
iN piN
p
seems to happen very rarely in dimension 4; examples have only very recently been found [Schi, Shi].
3 Motivation
The motivation for setting up this algorithm comes from [BS, SP]. In these articles we investigated linear
combinations of theta series of quaternary integral quadratic forms attached to ideals in definite quaternion
algebras. Recall that for an integral positive definite lattice (L, Q) as in Sect. 1 the theta series of degree r is
defined as
I:
iJ(r)(L, Q, Z) =
exp (27ri trace
((~B(;v;, ;~;j))Z)
(x1, ... ,x.)EL•
=
L
r(L, Q, T) exp (27ri trace TZ),
T~O
where Z E .fJr ={X +iY E M;Ym(C)IY pos. definite}, T E M:Ym(~1.) is positive semidifinite and r(L, Q, T) =
#{(x1, .. . ,xr) E FltB(xi,Xj) = T;j}.
In particular we could completely characterize the linear dependence relations between the theta series of degree
2 of the classes in the genus of even quaternary quadratic forms of level q and discriminant q2 for a prime q.
The first q for which such relations occur is q = 389.
140
H
L: ai19( 2)(£i, Qi, Z)
If
;:::: 0 is such
a
nontrivial linear relation then a theorem of Kitaoka [Ki] implies that
i=l
H
L: ai-,9( 3 )(£,, Q,, Z)
:/; 0 and one has reason to suspect that the Fourier coefficients of this modular form of
i=l
degree 3 satisfy interesting relations, possibly those of Yamazaki [Ya] for forms in the generalized MaaB space.
We have however not been able to prove that such relations do indeed hold.
To check for them, the first step obviously is to obtain a set of representatives of the classes in the genus, the
discriminant 389 2 being far out of the range of all existing tables. It will require further work to determine the
linear dependencies explicitly and check the Fourie:r coefficients in question. Algorithms for the computation of
these classes (more precisely, of the underlying ideals in the quaternion algebra considered) have been developed
by Pizer and by Shiota [Pi1, Shi], however, they produce along the way large numbers of useless lattices belonging
to other genera of other discriminants and levels. Our algorithm needed 41 seconds of CPU time on a decsystem
3100 workstation for the computation of all 319 dasses in this genus and 4 seconds for the determination of
a genus of 28 classes. It thus seems to behave roughly linear in the number of classes to be computed, which
is the expected value as long as all coefficients are in the range of single precision. One representative of each
genus described above can easily be written down as follows (so that we can start our algorithm) [Pi1, Prop.
5.2.]:
q:: 3 mod 4
q
(~
=5 mod 8
1
0
0
lli
lli
2
0
0
2
1
1
~)
1
fu2
2
2 4
q
0
q:: 1 mod 8
0
.ti:U
-q
~
2
-q
0
where pis a prime, p:: 3 mod 4, (~);:::: -1, a 2
::
6
~)
·-q mod p.
The number H of proper classes (i.e., classes up to isometry with determinant +1) in these genera can be
determined as follows:
The number h of classes of ideals in the definite quaternion algebra over Q of discriminant q that have a fixed
maximal left order is given by [Vi, p. 152]
h=
q -1
1
-4
1
-3
1:2 + 4(1 - ( q)) + 3(1- ( q )),
the number of types of maximal orders in that quaternion algebra by [Vi, p. 152]
1 [(1- (q-))
-4 + ,Bh(-q)]
t=2
where
,B =
1 q
2 q
{
4 q
=-1 mod 4
= 7 mod 8 or q = 3
= 3 mod 8, q :/; 3
and h( -q) is the number of ideal classes in the imaginary quadratic number field Q( A).
141
By [Pon] the number of proper classes in the genus is then given by t 2 + (h- t) 2 ; this can also be proved in
an elementary way using the description of isometries between ideal classes from [BS, Sect. 5] or [Shi]. In the
t 2 + (h- t) 2 + h
.
same way the number of classes is then seen to be
2
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